By hypothesis (ii) the merit function \(\nu _{+K}:\mathbb {X}\longrightarrow [0,+\infty ]\)  is l.s.c. on \(\mathbb {X}\)  and therefore the set
\([\nu _{+K}>0]\)  is open. By hypothesis (iii) it is \(x_0\in [\nu _{+K}<+\infty ]\ne \varnothing \) . Moreover, observe that
\((\mathbb {X},\Vert \cdot \Vert )\) , as a reflexive Banach space, is an Asplund
space. According to [1]},
this is equivalent to the fact that \((\mathbb {X},\Vert \cdot \Vert )\)  is
\(\widehat{\partial }\) -trustworthy. Such a property allows one to exploit
the slope estimate in [1]}
valid for l.s.c. functions on open subsets of \(\widehat{\partial }\) -trustworthy
Banach spaces, according to which
\( \inf _{x\in [\nu _{+K}>0]}|\nabla \nu _{+K}|(x)\ge \inf _{x\in [\nu _{+K}>0]}{\rm dist}\left(\mathbf {0}^*,\widehat{\partial }\nu _{+K}(x)\right).\) 
By recalling that for any \(x\in \mathbb {X}\)  the following general
relation between subdifferentials holds
\(\widehat{\partial }\nu _{+K}(x)\subseteq \partial _{\rm M}\nu _{+K}(x),\) 
from inequality \((\ref {in:stslFsubdest})\)  one obtains
\(\inf _{x\in [\nu _{+K}>0]}|\nabla \nu _{+K}|(x)\ge \inf _{x\in [\nu _{+K}>0]}{\rm dist}\left(\mathbf {0}^*,\partial _{\rm M}\nu _{+K}(x)\right).\) 
Now, as \(\nu \)  and \({\rm dist}\left(\cdot ,K\right)\)  are l.s.c. and Lipschitz
continuous on \(\mathbb {X}\) , respectively, they form a semi-Lipschitzian
sum at any \(x\in {\rm dom}\, \nu _{+K}\)  in the sense of [3]}.
Thus, as \((\mathbb {X},\Vert \cdot \Vert )\)  is an Asplund space, according
to the sum rule for the Mordukhovich subdifferential
(see [3]}) one finds
\(\partial _{\rm M}\nu _{+K}(x)\subseteq \partial _{\rm M}\nu (x)+\partial _{\rm M}{\rm dist}\left(\cdot ,K\right)(x)\subseteq \partial _{\rm M}\nu (x)+\widehat{B}^*_K(x),\) 
where the last inclusion holds because the Mordukhovich
subdifferential coincides with the Fenchel subdifferential,
in consideration of the convexity of \(K\) .
By virtue of hypothesis (iv), the last inclusion implies
\(\inf _{x\in [\nu _{+K}>0]}|\nabla \nu _{+K}|(x)\ge \gamma .\) 
Such an inequality enables one to apply Proposition REF ,
from which all the assertions in the thesis can be deduced.
This completes the proof.
